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Molecular Dynamics
Basics (4.1, 4.2, 4.3)
Liouville formulation (4.3.3)
Multiple timesteps (15.3)
Molecular Dynamics
• Theory:
2
d r
Fm 2
dt
• Compute the forces on the particles
• Solve the equations of motion
• Sample after some timesteps
2
3
4
5
6
Molecular Dynamics
• Initialization
– Total momentum should be zero (no external
forces)
– Temperature rescaling to desired temperature
– Particles start on a lattice
• Force calculations
–
–
–
–
Periodic boundary conditions
Order NxN algorithm,
Order N: neighbor lists, linked cell
Truncation and shift of the potential
• Integrating the equations of motion
– Velocity Verlet
– Kinetic energy
7
Periodic boundary conditions
8
Lennard Jones potentials
•The Lennard-Jones potential
12
6

 
  
LJ
u r   4      
 r  
 r 
•The truncated Lennard-Jones potential
u LJ r  r  rc
ur   
r  rc
 0
•The truncated and shifted Lennard-Jones potential
u LJ r   u LJ rc  r  rc
ur   
0
r  rc

9
Phase diagrams of Lennard Jones
fluids
10
Saving cpu time
Cell list
Verlet-list
11
Equations of motion
t 2
t 3
&
r t  t   r t   v t t 
f t  
r&&t    t 4
2m
3!
 
t 2
t 3
&
r t  t   r t   v t t 
f t  
r&&t    t 4
2m
3!
t 2
r t  t   r t  t   2r t  
f t    t 4
m
 
 
Verlet algorithm
t 2
r t  t   2r t   r t  t  
f t 
m
Velocity Verlet algorithm
t 2
r t  t   r t   v t t 
f t 
2m
t
v t  t   v t  
 f t  t   f t 
2m
12
r  0 , p  0
r  0 , p 0 ,
N
Lyaponov instability
N
N
1
, p j  0    , pi  0   
, p N 0

  1010
13

Liouville formulation
N
f p ,r
N

f  r f  p f
r
p


iL  r  p
r
p
df
 iLf
dt
Solution
Depends implicitly on t
Beware: this solution
Liouville
operatoruseless as
is equally
the differential
equation!
f t   exp iLt  f  0
14


iL  iLr  iL p  r&  p&
r
p
f t   exp iLr t  f 0 


 exp  r&0 t  f 0 

r 


 exp  p&0 t  f 0 

p 
n
&
p
0
t






Taylor
expansion

f
0

f 0 



n
n
n! r
n!
p
n 0
n 0
N
N
N
N
&
&
 f p 0 , r 0   r 0 t   f p 0   p 0 t  ,r 0 

r&0t 
 
Let us look at them
f t   exp
iL p t f 0
separately
n

n

Shift of coordinates
r 0   r 0   r&0 t
n
 

Shift of momenta
p 0   p 0   p&0 t
15
iLr  r 0   r 0   r&0 t
iL p  p 0   p 0   p&0 t
 t ,re te e e
f p
N
 0,r 0

Trotter identity
e
f p 0,r 0
 e  e  e 
e
 lim
e e
f p 0 ,r 0 

e
 e e e
N noncommuting
iLt  operators!
N
We have
A B
f p
A B
A B
iLr t iL p t
P 
A 2P B / P
iLr t
iL p t
N
N
N
A 2P P
N
N
A 2P B / P A 2P P
A B
B iLr t
t
A iL p t

t 

Pt 2 P
PP
P
P
iL
iL
t
2
f p N t , r N t   e p eiLr t e p  f p N 0 , r N 0 






16

iL t
e p

iL t 2  iL t  iL t 2 

e
e
e
iLr t  r  r  r&t
&
iLp t  p  p  pt
p
r
p
N


t
2


N
N
N
f p 0 , r 0   f   p 0   p&0  , r 0 
2




P

N
N
N




t
t
t







Velocity
iLr t 
N Verlet! 
e
f   p 0   p&0  , r 0   f   p 0   p&0  ,  r 0   t&
r   
 2  
2
2

 

 1 
N2
Nt t 




r
t


t

r
t

v
F
(
t
)

t

t
iL p t 2 
 t   

 

mr      ....................
e
f   p 0   p&0  ,  r 0  2t&
 2  
2
 

t

vt  t   v t  
F (t )  F (Nt  t )
N
t
t
t
  

 
2
m
f p 0   p&0   p&t  , r 0   t&
r


2
2

t
p 0   p 0    p&0   p&t 
2


   
2  
t 2
r 0   r 0   t&
r t 2   r 0   t&
r 0  
F(0)
2m
17
Velocity Verlet:
iLp t 2 iLr t 
e
e
iLp t 2
e
Call force(fx)
Do while (t<tmax)
e
iLp t 2
 iLr t 
e
e
t
 t 
: v  t    v t  
f t 
2
2m

vx=vx+delt*fx/2
: r  t  t   r  t   tv  t  t 2 
iLp t 2
x=x+delt*vx
Call force(fx)
t
: v  t  t   v  t  t 2  
f  t  t 
2m
vx=vx+delt*fx/2
enddo
18
Liouville Formulation
Velocity Verlet algorithm:
t
p t   p t  t 
2
t 2
r t  t   r t   tr t  
F(t )
2m
pt  t   pt  
Three subsequent coordinate transformations in either r or p of
which the Jacobian is one: Area preserving
t
t
pt  t 2   pt   Fr  pt  t 2   pt  t 2 




p
t


t

p
t


t
2

Fr t 
2
2
t
r t  t   r t   pt  t 2
r t   r t 
r t   r t 
m
t Fr 
t Fr 
1
J1  det
2
0
r
1
1
1
0
J 2  det
1
t m 1
J 3  det
1
0
2
r
1
1
Other Trotter decompositions are possible!
19
Multiple time steps
• What to use for stiff potentials:
– Fixed bond-length: constraints (Shake)
– Very small time step
20
F  Fshort  Flong
 F 
iL  iLr  iL p  v 
r m v
iL p  iLshort  iLlong
iLshort
Fshort 

m v
Trotter expansion:


i Llong Lshort Lr t
e
Multiple
Time steps
Flong 
iLlong 
m v
iLlongt 2 i  Lshort Lr t iLlongt 2
e
e
e
Introduce: δt=Δt/n

e
  eiLlong t 2 eiLshort  t 2 eiLr t eiLshort  t 2 n eiLlong t 2


i Llong  Lshort  Lr t
21

e
  eiLlong t 2 eiLshort  t 2 eiLr t eiLshort  t 2 n eiLlong t 2


i Llong  Lshort  Lr t
iLlong t 2  v  v  Flong t 2m
iLshort  t 2  v  v  Fshort  t 2m
iLr t  r  r  v t
First
e
iLlong t 2
f  r 0 , v(0)   f  r 0 , v 0   Flong 0 t 2m 
Now n times:
 e
iLshort  t 2 iLr  t iLshort  t 2
e
e
 f  r 0 ,v 0   Flong 0 t 2m 
n
22
e
t
 t 
: v  t    v t  
flong  t 
2 
2m

 iLpLong t 2
iLlong t 2
iLshort  t 2 iLr  t iLshort  t 2
e
Call force(fx_long,f_short)
e
vx=vx+delt*fx_long/2
Do ddt=1,n
e
iLpShort t 2

e
e
n
 e
iLlong t 2
iLlong t 2  v  v  Flong t 2m
iLshort  t 2  v  v  Fshort  t 2m
 t  t 
 t   t
: v  t     v  t iLr t  r frShort
 vtt
2 2
2  2m


vx=vx+ddelt*fx_short/2
 iLr t 
e
: r  t   t   r  t    tv  t  t 2   t 2 
x=x+ddelt*vx
iLlong t 2
iLshort  t 2 iLr  t iLshort  t
Call force_short(fx_short)

e
e
e e
e
iLpShort
n
iLlong t 2

e


 t 2
 t

 t  t   t
: v  t    t   ivLlong
t t 2 v
 vF
fShort
t 2mt 
long 

2 2  2m

iLshort  t 2  v  v  Fshort  t 2m
vx=vx+ddelt*fx_short/2
23
enddo
iLr t  r  r  v t
2

2
24